22.4 Market Supply

Last Wednesday we analyzed the problem of a profit-maximizing firm that bought resources (labor and capital) and used those resources to produce goods, which it then sold. We defined a perfectly competitive firm as one which was a price taker in both input and output markets: specifically, we assumed that they could buy any quantity of labor and capital at prices $w$ and $r$ respectively, and they could sell any amount of output at price $p$.

Just as we wrote the quantity of a good demanded by an individual consumer as $d(p)$ and the market demand as $D(p)$, let’s write the amount of a good supplied by a firm as $s(p)$ and the market supply as $S(p)$. And just as we aggregated the individual demands of all consumers in a market to get $D(p)$, we need to aggregate the quantity supplied by all firms to determine the market supply curve $S(p)$. Thus, if we have $N_F$ firms, and $s^j(p)$ is the supply function for firm $j$, we can write the overall market supply $S(p)$ as \(S(p) = s^1(p) + s^2(p) + s^3( p) + \cdots + s^{N_F}(p)\) or, more succinctly, \(S(p) = \sum_ {j=1}^{N_F}s^j(p)\) which we can read as “the total quantity of a good supplied at price $p$ is the sum, for each $j$ from 1 to $N_F$, of the quantity supplied by each firm $j$ at that price.”

Just as we did with demand, let’s look at an example in which there are $N_F$ identical firms, just to keep the math simple. In this case, if each firm has the individual supply function $s(p)$, the total market supply is just the number of firms times the amount supplied by each firm: \(S(p) = \sum_{i = j}^{N_F}s^j(p) = N_Fs(p)\) For example, let’s suppose that in some market, there are $N_F$ firms who each have access to the Cobb-Douglas production function \(f(L,K) = \sqrt{LK}\) We showed in Chapter 13 that, if capital is fixed at $\overline K$, the short-run supply function for such a firm is \(s(p) = {\overline Kp \over 2w}\) Therefore the total amount supplied by the market is \(S(p) = N_Fs(p) = {N_F \overline Kp \over 2w}\) The following diagrams show this situation. The diagram on the left shows the supply curve for each firm; the diagram on the right shows the market supply.

See interactive graph online here.

Again, you can play with the sliders to see how the individual supply curve and market supply curves are affected.